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Multispecies approach implemented in the code is based on the paper "Multi-species Lattice Boltzmann Model and Practical Examples. Short Course material Pietro Asinari PhD."

Equlibrium distribution function for multispecies is given in the paper as $$f^{\sigma(eq)}_\alpha(\rho,\mathbf{u^*_{\sigma}}) = \omega_\alpha\cdot\Big[s^{\sigma}_{\alpha}+\frac{1}{c^2_s} (\mathbf{e}_\alpha \cdot\mathbf{u^*_{\sigma}})+\frac{1}{2c^4_s}(\mathbf{e}_\alpha \cdot\mathbf{u^*_{\sigma}})^2-\frac{1}{c^2_s}(\mathbf{u^*_{\sigma}}\cdot \mathbf{u^*_{\sigma}})\Big]$$

where, $s^{\sigma}_0 = (9-5 \phi^\sigma)/4$ $s^{\sigma}_{\alpha} = \phi^\sigma$ for $1\leq\alpha\leq8$ and $\phi^\sigma=min_\varsigma(m^\varsigma)/m^\sigma$, $p_\sigma=\rho_\sigma\phi^\sigma/3$ $m^\sigma$ is molecular weight for species $\sigma$.

$u^*_{\sigma}$ is given as $$\mathbf{u}^*_{\sigma} = \mathbf{u}_\sigma + \sum_{\varsigma} {\frac{m^2}{m^\sigma m^{\varsigma}}\frac{B_{\sigma \varsigma}}{B_{mm}} x_\varsigma (\mathbf{u}_\varsigma-\mathbf{u}_\sigma) }$$ $x_\sigma= \rho_\sigma/\rho$

Relaxation time is given as

$\lambda_\sigma = \frac{pB_{mm}}{\rho }$

Multispecies: Variable Transformation

A semi-implicit Lattice Boltzmann equation is given as,

$$f^{\sigma,+}(\mathbf{x}+\mathbf{e},t+1) = f^\sigma(\mathbf{x},t) + (1-\frac{1}{2}) \lambda^\sigma[f^{\sigma(eq)}-f^\sigma] + \frac{1}{2} \lambda{^\sigma,+}[f^{\sigma(eq),+}-f^{\sigma,+}]$$

Variable transformation presented in above paper involves three steps

Step 1. Transforming f -> g

$g^\sigma = f^\sigma-\frac{1}{2}\lambda^\sigma[f^{\sigma(eq)}-f^\sigma]$

Step 2. Stream and Collide i.e g -> g^+

$g^{\sigma,+} = g^\sigma + \frac{\lambda^\sigma}{1+\frac{1}{2}\lambda^\sigma} [f^{\sigma(eq)}-g^\sigma]$

Step 3. Back Transormation to f i.e g^+ -> f^+

$$f^{\sigma,+} = \frac{g^{\sigma,+} + \frac{1}{2}\lambda^{\sigma,+} f^{\sigma(eq),+}}{1+\frac{1}{2}\lambda^{\sigma,+}}$$

In back tranformation, to compute feq we need $\rho$ and $\mathbf{u}_\sigma$ $\rho$ can be computed directly from g $\rho^+_\sigma = <g^{\sigma,+}>$

where as the $\mathbf{u}_\sigma$ computed by solving the linear system of equation given below

$$<\mathbf{e}_i g^{\sigma,+}> = \Bigg[ 1 + \frac{1}{2} \lambda^{\sigma,+} \sum_\varsigma (\chi_{\sigma \varsigma} x^+_\varsigma) \Bigg] q^+_{\sigma,i} - \frac{1}{2} \lambda^{\sigma,+}x^+_\sigma \sum_\varsigma (\chi_{\sigma \varsigma}q^+_{\varsigma,i})$$

where, $\chi_{\sigma \varsigma}$ is, $$\chi_{\sigma \varsigma} = \frac{m^2}{m_\sigma m_\varsigma} \frac{B_{\sigma \varsigma}}{B_{\sigma \sigma}}$$

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